Equivariant cohomology for cyclic groups of square-free order

The main objective of this paper is to compute $RO(G)$-graded cohomology of $G$-orbits for the group $G=C_n$, where $n$ is a product of distinct primes. We compute these groups for the constant Mackey functor $\underline{Z}$ and for the Burnside ring Mackey functor $\underline{A}$. Among other things, we show that the groups $\underline{H}^\alpha_G(S^0)$ are mostly determined by the fixed point dimensions of the virtual representations $\alpha$, except in the case of $\underline{A}$ coefficients when the fixed point dimensions of $\alpha$ have many zeros. In the case of $\underline{Z}$ coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain $G$-complexes.